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A304564 Number of minimum total dominating sets in the n-triangular honeycomb bishop graph.

%I A304564 #18 May 16 2025 22:59:18
%S A304564 0,2,2,6,75,21,208,3950,540,11220,314880,25740,917280,36029700,
%T A304564 1965600,107100000,5627890800,219769200,16995484800,1153034190000,
%U A304564 33844456800,3525796058400,300234909744000,6868433880000,927359072640000,96883959332160000,1776393899280000,301733192320560000
%N A304564 Number of minimum total dominating sets in the n-triangular honeycomb bishop graph.
%H A304564 Andrew Howroyd, <a href="/A304564/b304564.txt">Table of n, a(n) for n = 1..500</a>
%H A304564 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumTotalDominatingSet.html">Minimum Total Dominating Set</a>.
%H A304564 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularHoneycombBishopGraph.html">Triangular Honeycomb Bishop Graph</a>.
%F A304564 From _Andrew Howroyd_, Apr 04 2025: (Start)
%F A304564 a(3*n) = A382777(n).
%F A304564 a(3*n+4) = Sum_{k=0..n} A382776(n,k)*(4*binomial(n+k+2,2) * binomial(2*n-k+2,2) + 2*binomial(n+k+3,3) * (2*n-k+1)).
%F A304564 See the PARI program for a(3*n+2). (End)
%o A304564 (PARI)
%o A304564 T(n, k)=binomial(2*n-k, k)*binomial(n+k, n-k)*(2*(n-k))!*(2*k)!/(2^n)
%o A304564 b1(n) = sum(k=0, n, T(n,k))
%o A304564 b2(n) = sum(k=0, n, T(n,k)*(2*binomial(n+k+3,3)*(2*n-k+1) + 4*binomial(n+k+2,2)*binomial(2*n-k+2,2)))
%o A304564 b3(n) = sum(k=0, n, T(n,k)*(n+k)*(n+k+1)*(7*n-2*k+5)/3)
%o A304564 b4(n) = sum(k=0, n, T(n,k)*(2*binomial(n+k+4,4)*(2*n-k+1) + 24*binomial(n+k+2,2)*binomial(2*n-k+3,3)))
%o A304564 b5(n) = sum(k=0, n, T(n,k)*(40*binomial(n+k+6,6)*binomial(2*n-k+2,2) + 240*binomial(n+k+5,5)*binomial(2*n-k+3,3) + 304*binomial(n+k+4,4)*binomial(2*n-k+4,4)))
%o A304564 a(n) = my(t=n\3); if(n%3==0, b1(t), if(n%3==1, b2(t-1), b1(t+1) + b3(t) + b4(t-1) + b5(t-2))) \\ _Andrew Howroyd_, Apr 09 2025
%Y A304564 Cf. A304553, A304558, A382776, A382777.
%K A304564 nonn
%O A304564 1,2
%A A304564 _Eric W. Weisstein_, May 14 2018
%E A304564 a(8)-a(10) from _Andrew Howroyd_, May 19 2018
%E A304564 a(11) onwards from _Andrew Howroyd_, May 16 2025